Mathematics
Obstructions to distinguished Riemannian metrics via Lorentzian geometry
Document Type
Article
Abstract
We approach the problem of finding obstructions to curvature distinguished Riemannian metrics by considering Lorentzian metrics to which they are dual in a suitable sense. Obstructions to the lat-ter then yield obstructions to the former. This framework applies both locally and globally, including to compact manifolds, and is sensitive to various aspects of curvature. Here we apply it in two different ways. First, by embedding a Riemannian manifold into a Lorentzian one and utilizing Penrose’s “plane wave limit,” we find necessary local conditions, in terms of the Hessian of just one function, for large classes of Riemannian metrics to contain within them those that have parallel Ricci tensor, or are Ricci-flat, or are locally symmetric. Second, by considering Riemannian metrics dual to constant curvature Lorentzian metrics via a type of Wick rotation, we are able to rule out the existence of a family of compact Riemannian manifolds (in all dimensions) that deviate from constant curvature in a precise sense.
Publication Title
Advances in Theoretical and Mathematical Physics
Publication Date
2023
Volume
27
Issue
7
First Page
2219
Last Page
2259
ISSN
1095-0761
DOI
10.4310/ATMP.2023.v27.n7.a5
Keywords
projective, proper action, manifold
Repository Citation
Aazami, Amir Babak, "Obstructions to distinguished Riemannian metrics via Lorentzian geometry" (2023). Mathematics. 21.
https://commons.clarku.edu/mathematics/21
